Question

lim x→0 ​ 1−2cosx+cos2x ​÷ x²​

Answer 1

Step-by-step explanation:

We have,

lim

x→0

x

2

1−cosx

cos2x

=lim

x→0

x

2

(1+cosx

cos2x

)

1−cos

2

xcos2x

=

2

1

lim

x→0

x

2

1−cos

2

xcos2x

=

2

1

lim

x→0

x

2

1−cos

2

x(2cos

2

x−1)

=

2

1

lim

x→0

x

2

1+cos

2

x−2cos

4

x

=

2

1

lim

x→0

x

2

(−2cos

2

x−1)(cos

2

x−1)

=

2

1

lim

x→0

x

2

(2cos

2

x+1)(1−cos

2

x)

=

2

1

lim

x→0

x

2

(2cos

2

x+1)sin

2

x

=

2

1

×3×1=

2

3

Hence, the answer is

2

3

.